Computational processes

Size: px

Start display at page:

Download "Computational processes"

David Small
5 years ago
Views:

1 Spring 2010 Computational processes in living cells Lecture 6: Gene assembly as a pointer reduction in MDS descriptors Vladimir Rogojin Department of IT, Abo Akademi

2 Model forming To formalize gene assembly we need two steps: Formalize the DNA molecules that constitute the micronuclear, intermediate, and the assembled gene Formalize the way these objects are processed in gene assembly In our formalization we will keep a minimum of information that is still able to represent the essentials of the gene structure and keep track of gene assembly WE FORMALIZE: MIC gene Intermediate gene MAC gene 2

3 Formalizing the genes Idea: Represent the gene structure through the sequence of MDSs and pointers Represent the gene assembly as a process of sorting of MDSs, of composing bigger MDSs from its smaller parts and of pointer removals 3

4 First level: genes as MDS descriptors Example: gene actin I in S.Nova. Formalization (π 3, π 4 ) (π 4, π 5 ) (π 6, π 7 ) (π 5, π 6 ) (π 7, π 8 ) (π 9, e) (π2, π3) (b, π 2 ) (π 8, π M3 I1 M4 I2 M6 I3 M5 I4 M7 I5 M9 I6 π 3 π 4 π 4 π 5 π 6 π 7 π 5 π 6 π 7 π 8 π 9 e π3 M2 π2 I7 b M1 π 2 I8π 8 M8 (π 3, π 8 ) (π 9, e) (π2, π3) (b, π 2 ) (π 8, π 9 ) M3 M4 M5 M6 M7 I5 M9 I6 π 3 π 8 π 8 e π3 M2 π2 I7 M1 I8 M8 b π 2 π 8 π 9 (b, e) b M1 M2 M3 M4 M5 M6 M7 M8 M9 e 4

5 Genes as MDS descriptors M3 M4 M6 M5 M7 M9 M2 M1 M8 MDS descriptor: (3,4)(4,5)(6,7)(5,6)(7,8)(9,e)(-3,-2)(b,2)(8,9) 5

6 Level 1 Abstraction from pointers Second level: genes as legal Real genes strings Generic formalism: MDS arrangements MDS M i,j Abstraction from real DNA sequences, IESs and MDS content Abstraction from order numbers Permutations (we will not consider them any further during the course) Level 2 MDS descriptors MDS (i,j+1) Abstraction from MDSs Strings MDS i j+1 6

7 Representing genes as legal strings Note the degree of simplification at this level: from the nucleotide sequence of the gene we only represent at this level the order in which pointers (but not markers) occur along the sequence M3 M4 M6 M5 M7 M9 M2 M1 M8 Arrangements: M 3 M 4 M 6 M 5 M 7 M 9 M 2 M 1 M 8 MDS descriptor: (3,4)(4,5)(6,7)(5,6)(7,8)(9,e)(3,2)(b,2)(8,9) Legal string: denoted also as

8 Level 1 Third level: genes as signed graphs Real genes Abstraction from pointers Generic formalism: MDS arrangements MDS M i,j Abstraction from real DNA sequences, IESs and MDS content Abstraction from order numbers Permutations (we will not consider them any further during the course) MDS descriptors MDS (i,j+1) Level 2 Abstraction from MDSs Strings MDS i j+1 Abstraction from pointer sequence Level 3 Graphs 2 overlapping pointers p,q an edge between p,q 8

9 Signed overlap graphs: Example M3 M4 M6 M5 M7 M9 M2 M1 M8 Generic: M 3 M 4 M 6 M 5 M 7 M 9 M 2 M 1 M 8 MDS descriptor: (3,4)(4,5)(6,7)(5,6)(7,8)(9,e)(3,2)(b,2)(8,9) Legal string: denoted also as Overlap graph: 9

10 First level: MDS descriptors This is already a big abstraction from the real gene structure At this point we completely ignore the IESs and the bodies of the MDSs, i.e., most of the nucleotide sequence of the micronuclear gene However, the information that we do keep is sufficient to keep track of all intermediary structures produced in the gene assemble and to keep track of the assembly itself One more simplification: the real DNA sequence of each pointer is not important as long as we know their positions on the gene What is essential and we must keep is that the nucleotide sequence that is the incoming pointer of MDS i is identical with the nucleotide sequence making the outcoming pointer of MDS i-1 Remember also that MDS 1 starts with a beginning marker and the last MDS ends with an ending marker 10

11 First level: MDS descriptors Idea: denote each pointer by an integer: MDS i - (i,i+1), first MDS - (b,2), last MDS - (k,e) Note: the integers are used here just as a notation for a DNA sequence Example: M 3 is ACTGTTTAAA TATAATCGTA M 4 is CGTATAATA AATCTAGAGG MDS M 3 (3,4), where 3 stands for ACTG and 4 stands for CGTA MDS M 4 (4,5), where 4 stands for CGTA and 5 stands for CTAGAGG 11

12 Formalizing the process Formalize these intuitive ideas: MDS descriptors Next step: translate the molecular operation to MDS descriptors Based on this formal system prove the universality result: Any micronuclear gene can be assembled by a sequences of ld, hi, and dlad. 12

13 Mathematical preliminaries Alphabet: set of letters String over alphabet A: any sequence of letters from A Length of the string u: the number of letters in u Empty string: string of length 0 v is a substring of u if u=w 1 vw 2, v is proper if different from u and the empty string v is a permutation of u if u=a 1 a 2 a n, with a i A and v=a i1 a i2 a in, where (i 1,i 2,,i n ) is a permutation of (1,2,,n) 13

14 Mathematical preliminaries For an alphabet A, consider its signed copy A={a a A} A signed string over A is a string over A A Convention: a=a We say that a letter a A A occurs in the signed string u if either a or a occurs in u dom(u) A is the set of (unsigned) letters that occur in u if u=a 1 a 2 a n, a i A A, then its inversion is u=a n a 2 a 1, where a=a v is a signed permutation of u=a 1 a 2 a n, a i A if v=a i1 a i2 a in, where (i 1,i 2,,i n ) is a permutation of (1,2,,n) and a k {a k,a k } 14

15 MDS arrangements Step 1: drop the IESs, denote the MDSs as M i,j : Assume throughout this lecture that we are working on a gene having k MDSs in its micronuclear version; k is fixed for this lecture We use the alphabet Θ k ={ M i,j 1< i j < k } U { M 1,j 1 j < k } U { M i,k 1 < j k } U { M 1,k }. Letters M i,i are also denoted as M i Example: M 1, M 1,2, M 3, M 4,k letters from Θ k M 1, M 1,2, M 3, M 4,k letters from Θ k M 3,1,M 4,3 - not in Θ k 15

16 MDS arrangements Any signed string over Θ k is called an MDS arrangement We say that a signed string u over Θ k is an orthodox MDS arrangement if it is of the form u=m 1,i2-1 M i2,i3-1 M in,k Example: M 1,4 M 5,9 M 10 Any signed permutation of u is called a realistic arrangement Any signed permutation of M 1 M 2 M k is called a micronuclear arrangement 16

17 Generic formalism: Example The actin I gene in S.nova can be represented as M 3 M 4 M 6 M 5 M 7 M 9 M 2 M 1 M 8 by omitting all IESs Actin I in S.trifallax: M 3 M 4 M 6 M 5 M 7 M 9 M 10 M 2 M 1 M 8 17

18 Generic formalism: Example M 1,4 M 5,5 M 6,8 is an orthodox MDS sequence M 1,4 and M 6,8 are composite MDSs, M 5,5 is elementary and can be denoted also as M 5 M 3,5 M 9,11 M 1,2 M 12 M 6,8 is realistic It is a signed permutation of the orthodox MDS sequence M 1,2 M 3,5 M 6,8 M 9,11 M 12 M 1 M 1 is not a realistic MDS sequence 18

19 MDS descriptors Step 2 of the formalization Keep each MDS by its pair of pointers/markers; denote each pointer by an integer Alphabets: k ={2, 3,, k}, k ={2, 3,, k} Π k = k k Ψ={b,e,b,e} the set of markers Note: each integer and marker is a notation for a DNA sequence For any pointer p Π k, its pointer set is {p,p} 19

20 From generic formalism to MDS descriptors Represent each MDS by its pair of pointers each MDS M i will be represented as (i,i+1) and its inversion M i as (i+1,i) The first and the last MDSs are special represent M 1 as (b,2), M 1 as (2,b), M k as (k,e), and M k as (e,k), where b/e are special beginning/ ending markers This leads to the so-called MDS-descriptors MDS arrangement: M 3 M 4 M 6 M 5 M 7 M 9 M 2 M 1 M 8 MDS descriptor: (3,4)(4,5)(6,7)(5,6)(7,8)(9,e)(3,2)(b,2)(8,9) 20

21 MDS descriptors We work now with strings over the following alphabet: Γ k = { (b,e), (b,i), (i,e) 2 i k } { (i,j) 2 i < j k } Any signed string over the alphabet Γ k is called an MDS descriptor Example: (b,2)(e,b)(2,3) is an MDS descriptor Example: (b,2)(b,3)(b,4) is also an MDS descriptor We need to consider only realistic MDS descriptors 21

22 Denote the mapping bellow by ψ From generic formalism to MDS descriptors M i,j (i,j+1), 1<i j<k M i,j (j+1,i), 1<i j<k M 1,j (b,j+1), 1 j<k M 1,j (j+1,b), 1 j<k M i,k (i,e), 1<i k M i,k (e,i), 1<i k M 1,k (b,e) M 1,k (e,b) An MDS descriptor is realistic if it is the image through ψ of a realistic MDS arrangement 22

23 MDS descriptors: Example The realistic MDS descriptor associated to the actin I gene in O.nova is (3,4)(4,5)(6,7)(5,6)(7,8)(9,e) (-3,-2)(b,2)(8,9) The realistic MDS descriptor associated to the realistic MDS arrangement M 3,5 M 9,11 M 1,2 M 12 M 6,8 is: (3,6)(-12,-9)(b,3)(12,e)(-9,-6) 23

24 Realistic MDS descriptors Let δ=(x 1,x 2 )(x 3,x 4 ) (x 2n-1,x 2n ) be a realistic MDS descriptor: (x 2i- 1,x 2i ) k k We say that x 2i-1 is a left occurrence in δ and x 2i is a right occurrence in δ A pointer p occurs in δ if there is i such that x i =p Note: For an MDS descriptor δ and a pair (p,q) occurring in δ, we have: p k { b,e } if and only if q k { b,e } p k { b,e } if and only if q k { b,e } Let (p,q) Γ k If (p,q) occurs in δ, then p is called incoming and q is called outgoing If (-q,-p) occurs in δ, then p is called incoming and q is called outgoing 24

25 Realistic MDS descriptors Theorem: For a realistic MDS descriptor δ and a pointer p, δ has either 0 or 2 occurrences from the pointer set {p,-p} If both p and -p occur in δ, then p is called a positive pointer: here, either some pairs (p,q), (-p,-r), or some pairs (r,p), (-q,-p) occur in δ Otherwise (either p, or -p occur twice in δ), p is called a negative pointer: here, either some pairs (p,q), (r,p), or some pairs (-p,-r), (-q,-p) occur in δ 25

26 Realistic MDS descriptors Theorem: Let δ be a realistic MDS descriptor. Then: δ has exactly one occurrence form {b,-b} and one from {e,-e} Each pointer p has either 0 or 2 occurrences in δ A negative pointer has one left and one right occurrence in δ A positive pointer has either two left, or two right occurrences in δ If δ= δ 1 (x,y)δ 2, with (x,y) {(b,e),(-e,-b)}, then δ 1 = δ 2 =Λ 26

27 Take each of the operations Translating the molecular operations to realistic MDS descriptors Discuss for each of the operation the situations when the pointers involved are incoming or outgoing 27

28 The three molecular operations: LD, HI, DLAD (fold and recombine) LD (one piece of DNA is excised) HI (one piece of DNA is inverted) DLAD (two pieces of DNA exchange places) 28

29 Ld on realistic MDS descriptors If ld is applied in a successful assembly, then it must always be simple, so that no MDS is lost Simple ld: between the two occurrences of the pointer there are either no MDSs (just one IES), or the whole gene In the first case: excise a circular molecule consisting of one IES only In the second case: boundary application of ld the circular molecule contains the whole gene; in this case the gene will be assembled on a circular molecule 29

30 Ld on realistic MDS descriptors Case 1: simple ld one IES only separates the two occurrences of p Case 2: boundary ld the gene is bounded by the two occurrences of p 30

31 Formalizing the operations for MDS descriptors: LD ld p ( δ1 (q,p) (p,r) δ2 ) = δ1 (q,r) δ2 ld p ( (p,r) δ (s,p) ) = (s,r) δ (linear molecule) (circular molecule) LD: 31

32 Hi on realistic MDS descriptors Case 1: the first occurrence of p is as an incoming pointer (and then, also the second one) Case 2: the first occurrence of p is as an outgoing pointer (and then, also the second one) Result: the DNA sequence between the two pointers gets reverted 32

33 ld p ( δ1 (q,p) (p,r) δ2 ) = δ1 (q,r) δ2 ld p ( (p,r) δ (s,p) ) = (s,r) δ hi p ( δ1 (p,q) δ2 (-p,-r) δ3 ) = δ1 rs(δ2) (-q,-r) δ3 hi p ( δ1 (q,p) δ2 (-r,-p) δ3 ) = δ1 (q,r) rs(δ2) δ3 Formalizing the operations for MDS descriptors: HI HI: 33

34 Dlad on realistic MDS descriptors Four main cases: the first occurrence of p is incoming/outgoing and the first occurrence of q is incoming/outgoing Result: the two sequences of DNA bounded by p and q will change places 34

35 Dlad on MDS descriptors 1. The first occurrence of p is incoming, the first occurrence of q is incoming 2. The first occurrence of p is incoming, the first occurrence of q is outgoing 3. The first occurrence of p is outgoing, the first occurrence of q is incoming 4. The first occurrence of p is outgoing, the first occurrence of q is outgoing 35

36 Dlad on MDS descriptors 1. The first occurrence of p is incoming, the first occurrence of q is incoming 2. The first occurrence of p is incoming, the first occurrence of q is outgoing 3. The first occurrence of p is outgoing, the first occurrence of q is incoming 4. The first occurrence of p is outgoing, the first occurrence of q is outgoing 36

37 Dlad on MDS descriptors 1. dlad p,q (δ 1 (p,r 1 )δ 2 (q,r 2 )δ 3 (r 3,p)δ 4 (r 4,q)δ 5 ) = δ 1 δ 4 (r 4,r 2 )δ 3 (r 3,r 1 )δ 2 δ 5 2. dlad p,q (δ 1 (p,r 1 )δ 2 (r 2,q)δ 3 (r 3,p)δ 4 (q,r 4 )δ 5 ) = δ 1 δ 4 δ 3 (r 3,r 1 )δ 2 (r 2,r 4 )δ 5 3. dlad p,q (δ 1 (r 1,p)δ 2 (q,r 2 )δ 3 (p,r 3 )δ 4 (r 4,q)δ 5 ) = δ 1 (r 1,r 3 )δ 4 (r 4,r 2 )δ 3 δ 2 δ 5 4. dlad p,q (δ 1 (r 1,p)δ 2 (r 2,q)δ 3 (p,r 3 )δ 4 (q,r 4 )δ 5 ) = δ 1 (r 1,r 3 )δ 4 δ 3 δ 2 (r 2,r 4 )δ 5 37

38 Dlad on MDS descriptors 5. dlad p,q (δ 1 (p,r 1 )δ 2 (q, p)δ 4 (r 4,q)δ 5 ) = δ 1 δ 4 (r 4,r 1 )δ 2 δ 5 6. dlad p,q (δ 1 (p,q)δ 3 (r 3,p)δ 4 (q,r 4 )δ 5 ) = δ 1 δ 4 δ 3 (r 3,r 4 )δ 5 7. dlad p,q (δ 1 (r 1,p)δ 2 (q,r 2 )δ 3 (p,q)δ 5 ) = δ 1 (r 1,r 2 )δ 3 δ 2 δ 5 38

39 Dlad on realistic MDS descriptors 1. dlad p,q (δ 1 (p,r 1 )δ 2 (q,r 2 )δ 3 (r 3,p)δ 4 (r 4,q)δ 5 ) = δ 1 δ 4 (r 4,r 2 )δ 3 (r 3,r 1 )δ 2 δ 5 2. dlad p,q (δ 1 (p,r 1 )δ 2 (r 2,q)δ 3 (r 3,p)δ 4 (q,r 4 )δ 5 ) = δ 1 δ 4 δ 3 (r 3,r 1 )δ 2 (r 2,r 4 )δ 5 3. dlad p,q (δ 1 (r 1,p)δ 2 (q,r 2 )δ 3 (p,r 3 )δ 4 (r 4,q)δ 5 ) = δ 1 (r 1,r 3 )δ 4 (r 4,r 2 )δ 3 δ 2 δ 5 4. dlad p,q (δ 1 (r 1,p)δ 2 (r 2,q)δ 3 (p,r 3 )δ 4 (q,r 4 )δ 5 ) = δ 1 (r 1,r 3 )δ 4 δ 3 δ 2 (r 2,r 4 )δ 5 5. dlad p,q (δ 1 (p,r 1 )δ 2 (q, p)δ 4 (r 4,q)δ 5 ) = δ 1 δ 4 (r 4,r 1 )δ 2 δ 5 6. dlad p,q (δδ 1 (p,q)δδ 3 (r 3,p)δδ 4 (q,r 4 )δδ 5 ) = δ 1 δ 4 δ 3 (r 3,r 4 )δδ 5 7. dlad p,q (δ 1 (r 1,p)δ 2 (q,r 2 )δ 3 (p,q)δ 5 ) = δ 1 (r 1,r 2 )δ 3 δ 2 δ 5 39

40 Gene assembly as a transformation of MDS descriptors ld p ( δ1 (q,p) (p,r) δ2 ) = δ1 (q,r) δ2 ld p ( (p,r) δ (s,p) ) = (s,r) δ hi p ( δ1 (p,q) δ2 (-p,r) δ3 ) = δ1 rs(δ2) (q,r) δ3 hi p ( δ1 (q,p) δ2 (r,-p) δ3 ) = δ1 (q,r) rs(δ2) δ3 dlad p,q (δ1(p,r1)δ2(q,r2)δ3(r3,p)δ4(r4,q)δ5) = δ1δ4(r4,r2)δ3(r3,r1)δ2δ5 dlad p,q (δ1(p,r1)δ2(r2,q)δ3(r3,p)δ4(q,r4)δ5) = δ1δ4δ3(r3,r1)δ2(r2,r4)δ5 dlad p,q (δ1(r1,p)δ2(q,r2)δ3(p,r3)δ4(r4,q)δ5) = δ1(r1,r3)δ4(r4,r2)δ3δ2δ5 dlad p,q (δ1(r1,p)δ2(r2,q)δ3(p,r3)δ4(q,r4)δ5) = δ1(r1,r3)δ4δ3δ2(r2,r4)δ5 dlad p,q (δ1(p,r1)δ2(q,p) δ4(r4,q)δ5) = δ1δ4(r4,r1) δ2δ5 dlad p,q (δ1(p, q)δ3(r3,p)δ4(q,r4)δ5) = δ1δ4δ3(r3,r4) δ5 dlad p,q (δ1(r1,p)δ2(q,r2)δ3(p, q)δ5) = δ1(r1,r2)δ3δ2δ5 40

41 Assembly strategies A composition φ of operations ld, hi, and dlad is an assembly strategy for the MDS descriptor u if φ(u) is an assembled MDS descriptor (i.e. (b,e) or (-e,-b)) 41

42 The MDS descriptor associated to the actin I gene in S.nova is δ=(3,4)(4,5)(6,7)(5,6)(7,8) (9,e)(-3,-2)(b,2)(8,9) Ld 4 is applicable to δ: Ld 4 (δ)=(3,5)(6,7)(5,6)(7,8)(9,e)(-3,-2)(b,2)(8,9) Hi -2 is applicable to δ: Hi -2 (δ)=(3,4)(4,5)(6,7)(5,6)(7,8) (9,e)(-3,-b)(8,9) Hi 3 is applicable to δ: Hi 3 (δ)=(-e,-9)(-8,-7)(-6,-5)(-7,-6)(-5,-4)(-4,-2)(b,2)(8,9) Dlad 5,6 is applicable to δ: Dlad 5,6 (δ)=(3,4)(4,7)(7,8)(9,e)(-3,-2)(b,2)(8,9) Example 42

43 Example δ=(4,5)(6,e)(-2,-b)(5,6)(-4,-3)(-3,-2) dlad 5,-2 is applicable to δ: 43

44 Example δ=(4,5)(6,e)(-2,-b)(5,6)(-4,-3)(-3,-2) dlad 5,-2 is applicable to δ: dlad 5,-2 (δ)=(4,6)(-4,-3)(-3,-b)(6,e) 44

45 Example Actin I gene in S.nova is δ=(3,4)(4,5)(6,7)(5,6)(7,8)(9,e)(-3,-2)(b,2)(8,9) A successful reduction of δ is: Ld 4 (δ)=(3,5)(6,7)(5,6)(7,8)(9,e)(-3,-2)(b,2)(8,9) Dlad 5,6 (ld 4 (δ))=(3,7)(7,8)(9,e)(-3,-2)(b,2)(8,9) Ld 7(dlad 5,6(ld 4(δ)))=(3,8)(9,e)(-3,-2)(b,2)(8,9) δ Dlad 8,9 (ld 7 (dlad 5,6 (ld 4 (δ))))=(3,e)(-3,-2)(b,2) Hi -2 (dlad 8,9 (ld 7 (dlad 5,6 (ld 4 (δ)))))=(3,e)(-3,-b) Hi 3 (hi 2 (dlad 8,9 (ld 7 (dlad 5,6 (ld 4 (δ))))))=(-e,-b) 45

46 Translating the operations to realistic MDS descriptors From molecular operations based on patterns, folding, and splicing we went to a formal calculus on pairs of letters The gene assembly process is now a formal rewriting process based on three rules The input of the process: a realistic MDS descriptor The output of the process: the sequence of rewriting rules that transforms the input into (b,e) or (e,b); any such sequence is called a successful reduction Advantage: one can reason in formal terms about the process of gene assembly Note: Although the IESs are ignored in this rewriting calculus, note that they can be easily considered in the operations In dealing with certain applications (invariants) we will redefine the operations to keep track of the IESs also 46

47 Universality result Universality: Any realistic MDS descriptor has a successful reduction Consequence: Any micronuclear ciliate gene can be successfully assembled using a sequence of ld, hi, and dlad 47

48 Proof: Universality result For any realistic MDS descriptor δ, there is an operation applicable to it (thus reducing its length) If δ has a positive pointer, then apply hi to δ on that pointer; otherwise, all pointers are negative If δ has any alternating direct repeat pattern ( p q p q ), then apply dlad on p an q Otherwise, consider a (negative) pointer p of δ such that the distance (in number of pointers) between the two occurrences of p in δ is minimal Then the distance must be 0 and so, ld p is applicable to δ 48

49 Real genes Summary Level 1 Abstraction from pointers Generic formalism: MDS arrangements MDS M i,j Abstraction from real DNA sequences, IESs and MDS content Abstraction from order numbers Permutations (we will not consider them any further during the course) Level 2 MDS descriptors MDS (i,j+1) Abstraction from MDSs Strings MDS i j+1 49

COMPUTATIONAL PROCESSES IN LIVING CELLS

COMPUTATIONAL PROCESSES IN LIVING CELLS Lecture 7: Formal Systems for Gene Assembly in Ciliates: the String Pointer Reduction System March 31, 2010 MDS-descriptors MDS descriptors: strings over the following